How this square root calculator works
The square root of a number n, written √n, is the value that produces n when multiplied by itself. Because both a positive and a negative value square to the same result, every positive number has two square roots: the principal (non-negative) root and its negative. We report the principal root to several decimals, then the complete ± form.
The calculator also checks whether your number is a perfect square — a whole number whose root is itself a whole number, like 1, 4, 9, 16 or 144. When it is not, the tool gives the simplified radical form by factoring out the largest perfect square. As a bonus it shows the cube root (∛n, the value cubed to give n) and the square (n²) for reference.
Perfect squares reference (1–15)
Reference note: irrational roots such as √2 ≈ 1.41421356 never terminate or repeat, so decimal results are rounded. Negative inputs are reported in imaginary form, for example √−4 = 2i, and the square root of 0 is 0.
Frequently asked questions
- What is a square root?
- The square root of a number n is the value that, when multiplied by itself, gives n. It is written √n — for example √9 = 3 because 3 × 3 = 9. Every positive number has a positive (principal) root and a negative one, so the full answer is ±3.
- How do I simplify a square root?
- Factor the number into a largest perfect square times a remainder, then take the root of the perfect square outside the radical. For example √72 = √(36 × 2) = 6√2. If no perfect-square factor larger than 1 exists, it is already in simplest form.
- What is a perfect square?
- A perfect square is a whole number that is the square of an integer — such as 1, 4, 9, 16, 25 or 144. Its square root is itself a whole number, so √144 = 12 exactly with no decimal part.
- Can you take the square root of a negative number?
- Not within the real numbers, because no real value squared gives a negative result. The answer is imaginary: √−n = √n × i, where i² = −1. For example √−4 = 2i.
- What is the square root of 2?
- The square root of 2 is an irrational number, approximately 1.41421356. It cannot be written exactly as a fraction or terminating decimal, so it is usually left as √2 or rounded.
- How is a square root different from a cube root?
- A square root asks which value squared gives n (√n), while a cube root asks which value cubed gives n (∛n). For example √64 = 8 because 8² = 64, but ∛64 = 4 because 4³ = 64. Cube roots are also defined for negatives, since ∛−8 = −2.